
Note that, as before, in the course of the implementing microscopic constraints, (REF ) and (REF ), where each one constrains densities both in \(R\)  and \(L\)  sector, the implied gauge fields connect i.e., transform, at the same time, in the \(R\)  and \(L\)  sectors of fields \(c\)  and \(d\) . We warn the reader that, as it stands, in the proposed covariant derivatives, only fields \(b^{(1)}_{\mu ,\pm }\equiv a_{\mu }^{(1)}\pm A_{\mu }\)  and \(b^{(2)}_{\mu ,\pm }\equiv a_{\mu }^{(2)}\pm A_{\mu }\)  are assumed to transform canonically (see Appendix B for details), and fields \(a_{\mu }^{(1)}\) , \(a_{\mu }^{(2)}\) , and \( A_{\mu }\) , that enter their decompositions (as they refer both to \(R\)  and \(L\)  sectors) do not. The complete set-up of the new NC theory is given in terms of \(b^{(1)}_{\mu ,\pm }\)  and \(b^{(2)}_{\mu ,\pm }\)  only. In the Appendix B we explain why necessarily \(b^{(1)}_{\mu ,\pm } = b^{(2)}_{\mu ,\pm }\)  = \(b_{\mu ,\pm }\) , i.e., \(a_{\mu }^{(1)} = a_{\mu }^{(2)}\) , and we have an emerging symmetry between the unphysical and physical sectors. The fields \(a_{\mu } = b_{\mu ,+} + b_{\mu ,-}\)  and \(A_{\mu } = (b_{\mu ,+} - b_{\mu ,-})/2 \)  will assume their expected \(U(1)\)  gauge field roles in the commutative limit of the new theory in which we get
the original version of the Son's theory that incorporates the PH symmetry (CP or CT in [1]}). The resulting NC action up to first order is given by
\(S_{NC}&=\int d\tau d^{2}x\Big ( \hat{c}^{\dagger } D_\tau \hat{c} + \hat{d}^{\dagger } D_\tau \hat{d}+\hat{c}^\dagger (iD_x +D_y)\hat{d}+ \hat{d}^\dagger (iD_x-D_y)\hat{c}+ i \hat{a}_{0} \bar{\rho }_e \Big )+\frac{i}{4\pi }\varepsilon ^{\alpha \beta \gamma }\int d\tau d^{2}x\;A_{\alpha }\partial _{\beta }a_{\gamma }\nonumber \\&+\theta \int d\tau d^{2}x\left(\hat{c}^{\dagger }\hat{F}_{12}D_{\tau }\hat{c}-\hat{c}^{\dagger }\hat{F}_{10}D_{y}\hat{c}+\hat{c}^{\dagger }\hat{F}_{20}D_{x}\hat{c}+\hat{d}^{\dagger }\hat{F}_{12}D_{\tau }\hat{d}-\hat{d}^{\dagger }\hat{F}_{10}D_{y}\hat{d}+\hat{d}^{\dagger }\hat{F}_{20}D_{x}\hat{d} \right).\) 
